Even if the semismooth Newton method enjoys a fast rate of convergence, computing the exact solution of (REF ) using a direct method can be expensive if the number of unknowns is large. A more computationally efficient solution in the large-scale case consists in solving (REF ) only approximately with some iterative linear solver and using a certain stopping rule. These are the principles behind inexact semismooth Newton methods [1]}, [2]}. In particular, \(V_{k+1}\)  is no longer required to exactly solve (REF ), but only to satisfy
\(\Vert r(V_k) + J_k \left( V_{k+1} - V_k \right) \Vert \le \alpha _k\Vert r(V_k) \Vert \) 
